Slideshow 10, MathematicsMr Richard Sasaki, Room 307

Powers and Indices

Objectives• To recall algebraic rules learned so far• To learn how products of an unknown

make a power• To learn how to multiply and divide

powers of an unknown

ReviewLet’s review the main rules we have learned so far.6×𝑥¿6 𝑥𝑥×𝑥¿𝑥2𝑥× 𝑦¿𝑥𝑦𝑥+𝑥¿2 𝑥𝑥+𝑦¿𝑥+𝑦

−5×𝑥¿−5 𝑥𝑥−𝑥¿0𝑥÷ 𝑦¿𝑥𝑦𝑥÷ 𝑥¿1

Also, writing expressions in alphabetical order is usually preferred () but not crucial. ( is fine).

Powers (Indices)As we know, . 𝑥2We call or .-squared to the power 2

The small 2 symbol at the top is called the power or index.Note: Power and Index mean the same thing. Indices is plural of index in this context.

How about ? 𝑥×𝑥×𝑥=𝑥3We call or .-cubed to the power 3How about ? 𝑥×𝑥×𝑥×𝑥=𝑥4

We call . to the power 4

Note: onwards are read “to the power” as well.

Calculation (Multiplication)What do you think is? 𝑥1=𝑥Just one is present.

Let’s try some multiplication.

ExampleCalculate . Have a guess!

𝑎4×𝑎3=¿(𝑎×𝑎×𝑎×𝑎)×(𝑎×𝑎×𝑎)¿𝑎7

So… .𝑥𝑎+𝑏

What will happen when we divide indices?

Note: Powers is one area where we see and symbols in algebra (before simplified).

Calculation (Division)ExampleCalculate .

𝑎6÷𝑎3=¿𝑎×𝑎×𝑎×𝑎×𝑎×𝑎

𝑎×𝑎×𝑎¿𝑎×𝑎×𝑎¿𝑎3

So… .𝑥𝑎−𝑏

What do you think might equal? Have a think!

Answers𝑥3 𝑦 5 𝑥5𝑎7 𝑥3 𝑦 6

𝑥2 𝑦 3 𝑥𝑎6 𝑎 𝑥6

𝑎5 𝑎7 𝑥6

𝑥3 𝑦 7 𝑥12

𝑥9 𝑦 4 𝑦 5

𝑎9 or

1

Negative Powers and Zero and . So , right? Why?

𝑥3÷ 𝑥5𝑥3−5𝑥− 2

𝑥3

𝑥5𝑥×𝑥×𝑥

𝑥×𝑥×𝑥×𝑥×𝑥1

𝑥2

So… .1

𝑥𝑎 Writing this in both ways is fine.

Why does ?

𝑥0=¿𝑥1÷𝑥1=¿𝑥÷ 𝑥=¿1Note: Any number to the power zero is 1.

Answers1𝑥

1

𝑦31

𝑦31

𝑎5

11 1 1

𝑥− 1 𝑥− 5 𝑥− 3 𝑥− 5

1

𝑥21

𝑎4

1

𝑦41𝑥

2𝑥

3

𝑦32

𝑎27

𝑥33

𝑎2𝑥2

𝑦

Brackets and Other CalculationsHow would we calculate ?

(𝑥2 )3=¿𝑥2×𝑥2×𝑥2=¿𝑥6So… .𝑥𝑎𝑏

Be careful! (usually).ExampleCalculate .

4 (𝑎2 )3×2𝑎2=¿4× (𝑎2 )3×2×𝑎2¿8×𝑎6×𝑎2¿8 𝑎8

Answers - Easy𝑥7 𝑥3 𝑥121 𝑎 𝑎2 𝑥 5 𝑦 3 52 3 𝑥2𝑦 2

0 𝑥6 𝑦 𝑥2

4 𝑥2 16 𝑦2 4 𝑥4

2 𝑥2 𝑎2𝑏2 6 𝑥3

2 𝑥3 6 𝑥3 𝑦 2 𝑥2 𝑦3

𝑥 1 𝑥− 3

𝑥 2 3

Answers - Hard

2 𝑥− 2 3 𝑦−3 3𝑎−2 2𝑎−3

0 8 𝑥6 21 𝑥6 𝑦6 𝑥7

3 𝑥 𝑦2 4 𝑥2 𝑦 2 3 𝑥2

𝑥 𝑦 312 2 𝑥6 𝑦3

8 𝑥6 𝑦3 256 𝑥12 2 𝑥3𝑥2 𝑥9 𝑦3 𝑥4 𝑥𝑦 96 𝑥5 𝑦10